We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in [2] to arbitrary semialgebraic sets.All previous algorithms proposed for this problem have doubly exponential complexity.

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Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, the worst-case analysis of root-finding algorithms does not correlate with their practical performance. We develop a smoothed analysis framework for polynomials with integer coefficients to bridge the gap between the complexity estimates and the practical performance. In this setting, we derive that the expected bit complexity of Descartes solver to isolate the real roots of a polynomial, with coefficients uniformly distributed, is O B (d 2 + dτ ), where d is the degree of the polynomial and τ the bitsize of the coefficients.

Solving polynomials is a fundamental computational problem in mathematics. In the real setting, we can use Descartes' rule of signs to efficiently isolate the real roots of a square-free real polynomial. In this paper, we translate this method into the padic worlds. We show how the p-adic analog of Descartes' rule of signs, Strassman's theorem, leads to an algorithm to isolate the roots of a square-free p-adic polynomial. Moreover, we show that this algorithm runs in O(d 2 log 3 d)-time for a random p-adic polynomial of degree d. To perform this analysis, we introduce the condition-based complexity framework from real/complex numerical algebraic geometry into p-adic numerical algebraic geometry.

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We present a version of Smale's α-theory for ultrametric fields, such as the p-adics and their extensions, which gives us a multivariate version of Hensel's lemma. Hensel's lemma [4, §3.4] gives us sufficient condition for lifting roots mod p k to roots in Z p . Alternatevely, Hensel's lemma gives us sufficient conditions for Newton's method convergence towards an approximate root. Unfortunately, in the multivariate setting, versions of Hensel's lemma are scarce [2]. However, in the real/complex world, Smale's α-theory [3] gives us a clean sufficient criterion for deciding if Newton's method will converge quadratically. In the p-adic setting, Breiding [1] proved a version of the γ-theorem, but he didn't provide a full α-theory. In this short communication, we provide an ultrametric version of Smale's α-theory for square systems-initially presented as an appendix in [5]-, together with an easy proof.In what follows, and for simplicity 1 , F is a non-archimedian complete field of characteristic zero with (ultrametric) absolute value | | and P n,d [n] the set of polynomial maps f : F n → F n where f i is of degree d i . In this setting, we will consider on F n the ultranorm given by x := max{x 1 , . . . , x n }, its associated distance dist(x, y) := x − y , and on k-multilinear maps A : (F n ) k → F q the induced ultranorm, which is given by

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